Question

State two conditions for the function f(x,y) to be continuous joint probability distribution of continuous random variables X and Y

Answer 1

If random variables $X$ and $Y$ are continuous and are defined on the same sample space $S$, then, their joint probability density function(pdf) is a continuous function given by,

$f(x,y)$. The function $f(x,y)$ must satisfy the following conditions.

• $f(x,y)\geqslant0,\forall(x,y)\in\R^2$
• $\displaystyle\iint_{\R^2}f(x,y)dxdy=1$

where $\R^2$ is the set of pairs of real numbers given as,

$\R^2=\R*\R=\{(x,y):x\space and \space y\space are\space real\space numbers\}$